QMGreensite_merged

244 CHAPTER15. IDENTICALPARTICLES

if(nlm)and(n′m′l′)aredifferent,and

Ψsingnlm,nlletm=φnlm(r 1 ,θ 1 ,φ 1 )φnlm(r 2 ,θ 2 ,φ 2 )Φ 00 (15.48)

if(nlm)and(n′m′l′)arethesame. Notethatinthegroundstate

Ψsing 100 , 100 let=φ 100 (1)φ 100 (2)×

1

√

2

[χ^1 +χ^2 −−χ^1 −χ^2 +] (15.49)

theelectronsareinspinup/spindowncombinations. Itisimpossibleto put two
spinupelectrons(ortwospindownelectrons)intotheHeliumgroundstate;thisis
anotherillustrationoftheexclusionprinciple.
Nowletsconsiderthecasewhereoneelectronisinthe(100)state,andtheother
electronisinthe(nlm)state. Thes= 0 singlet combinationisknownas Para-
helium,andthes= 1 tripletcombinationisknownasOrthohelium. Neglecting
theelectrostaticrepulsivepotentialV′betweenthetwoelectrons,Orthoheliumand
Paraheliumaredegenerateinenergy.
InordertoseehowtheelectronrepulsionremovesthedegeneracybetweenOrtho-
andParahelium,wewilltreatV′asthoughitwereasmallperturbingpotential,and
usetheformula
∆E=<Ψ|V′|Ψ> (15.50)

whichisvalidifV′isasmallcorrectiontotherestoftheHamiltonian. Actually,
inthiscaseV′isnotsosmall;itisthesameorderofmagnitudeinstrengthasthe
attractivepotentialoftheelectronstothenucleus. Butwewillgoaheadanduseeq.
(15.50)anyway,trustingthatitwillproduceresultsthatareatleastqualitatively,if
notquantitatively,correct. Sothecorrectiontothezerothorderenergy

EO,P=E 0 +∆E (15.51)

forOrtho-andParaheliumwillbe

∆EO,P =

1

2

∫

d^3 x 1 d^3 x 2 {φ∗ 100 (x 1 )φ∗nlm(x 2 )±φ∗ 100 (x 2 )φ∗nlm(x 1 )}

×

e^2

|%x 1 −%x 2 |

{φ 100 (x 1 )φnlm(x 2 )±φ 100 (x 2 )φnlm(x 1 )}

×<Φs=0, 1 |Φs=0, 1 > (15.52)

wheretheplussignisforParahelium,andtheminussignforOrthohelium.Thespin
statesarenormalized,so<Φ|Φ>=1,andtherefore,collectingliketerms,

∆EP = A+B

∆EO = A−B (15.53)

where

A=

∫

d^3 x 1 d^3 x 2 |φ 100 (x 1 )|^2

e^2

r 12

|φnlm(x 2 )|^2 (15.54)